The MFS versus the Trefftz method for the Laplace equation in 3D

Lv, Hui; Fang, Hao; Yong, Wang; Chen, Ching-Shyang

doi:10.1016/j.enganabound.2017.06.006

Cited by 15 publications

(5 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After adopting the characteristic length in our numerical model, the ill-posed phenomenon is greatly reduced, and the accurate numerical solutions can be obtained. Collocating the numerical expansion from Equations (32) and (34) at boundary collocation points to match the given boundary conditions, we may obtain the following equation.…”

Section: The Characteristic Lengthmentioning

confidence: 99%

“…The CTM requires the evaluation of the coefficients in which they may be obtained by solving the linear simultaneous equations assembled by using the boundary conditions at a number of collocation points. Applications of the CTM such as Laplace and modified Helmholtz equations [32,33] and the problem of boundary detection [34] has been studied. Due to the complexity, applications of the CTM are most limited to the homogeneous problems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Xiao

Liu

2019

Water

View full text Add to dashboard Cite

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

show abstract

Section: The Characteristic Lengthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Xiao

Liu

2019

Water

View full text Add to dashboard Cite

show abstract

“…The Trefftz method can quickly solve the boundary value problem. At present, it has been applied in many fields of physics, mathematics, and engineering, the approximate solution can be expressed as a linear combination of functions that satisfy the control equation [9][10][11][12][13][14][15][16][17][18]. Li et al (2007,2008) made a comprehensive comparison of Trefftz and other boundary methods to conclude that Trefftz is the simplest method of calculation, providing the most accurate equation solution and optimal numerical stability [19][20].…”

Section: Introductionmentioning

confidence: 99%

Trefftz Method Applied to the Problem of Pumping Well Test Counter-Calculation

Su¹,

Huang²,

Yang³

et al. 2022

Pol. J. Environ. Stud.

View full text Add to dashboard Cite

Nowadays, field pumping test is still the primary way to obtain accurate hydrogeological parameters, and high cost and low efficiency are still its main limitations. Therefore, it is still beneficial to study how to obtain hydrogeological parameters quickly and accurately. In this paper, the Trefftz method is applied in combination with the New Fictitious Time Integration Method(NFTIM) , to solve the groundwater reverse calculation problem with unknown partial boundary and effectively solve the discomfort in the reverse calculation problem. Based on the actual pumping test data of the ZK03 hole in the section V of Harbin Metro Line 3, the model was built. By verifying the analytical solution with the Thiem formula and comparing it with the observation well data, it is proved that the Trefftz inverse calculation method is more operable and has a high degree of accuracy (with average error of 9.94 × 10 -10 ). Concurrently, the hydrogeological parameters such as the influence radius, water head distribution, and depth drop curve of the pumping well are calculated to provide crucial for the design and construction of subway engineering.

show abstract

“…In the past, many heat conduction problems in layered composites have been routinely solved by numerical methods such as the finite difference method [4][5][6], finite element method [7,8], finite volume method [9][10][11] and the boundary element method [12,13]. In contrast to mesh-based numerical methods, the meshless methods, which do not need the mesh generation and boundary integral have been proposed such as the analytical method [14], method of fundamental solutions (MFS) [15][16][17], boundary knot method (BKM) [18], collocation Trefftz method (CTM) [19][20][21], radial basis function collocation method (RBFCM) [22][23][24][25], element-free Galerkin method (EFG) [26], reproducing kernel particle method (RKPM) [27,28], modified polynomial expansion method [29], meshless local boundary integral equation method (LBIE) [30,31], and so on. Among these, boundary-type meshless methods have attracted considerable attention because of their simplicity.…”

Section: Introductionmentioning

confidence: 99%

A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

Xiao

Huang

et al. 2018

Applied Sciences

View full text Add to dashboard Cite

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.

show abstract

The MFS versus the Trefftz method for the Laplace equation in 3D

Cited by 15 publications

References 10 publications

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Trefftz Method Applied to the Problem of Pumping Well Test Counter-Calculation

A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

Contact Info

Product

Resources

About